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Doubly Reflected BSDEs in the predictable setting

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Pith's one-line read Doubly reflected BSDEs with predictable barriers have unique solutions

desk verdict The main theorem is false as stated because B_T and B′_T are unconstrained, but a one-line normalization fix restores it; the paper fills a genuine gap in the predictable-barrier DRBSDE literature. read the letter →

arxiv 1908.08076 v2 pith:5GJOSGOV submitted 2019-08-21 math.PR

Ihsan Arharas , Siham Bouhadou , Youssef Ouknine This is my paper · ORCID
classification math.PR MSC 60H2060H3065C30
keywords predictableDRBSDEsdoublyreflectedbackwardstochasticdifferentialequationsnonquasi-leftcontinuousfiltrationsMokobodzkiconditionPicarditerationmethodBanachfixedpointtheoremstrongsupermartingalesSkorohodconditions
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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes existence and uniqueness for doubly reflected backward stochastic differential equations (DRBSDEs) when the two reflecting barriers are predictable processes and the filtration need not be quasi-left-continuous. In that setting martingales can jump at predictable times, so the solution must be kept between the barriers by four increasing processes: two acting at left jumps and two at right jumps. The paper shows that the natural analogue of Mokobodzki's condition, namely the existence of two nonnegative predictable strong supermartingales whose difference lies between the barriers, is necessary and sufficient for a solution. The proof first solves the case where the driver does not depend on the solution by rewriting the problem as a coupled pair of one-barrier predictable reflected BSDEs and applying a Picard iteration, then obtains the general Lipschitz-driver case as the fixed point of a contraction. Because the filtration is general, the argument uses the Gal'chouk–Lenglart change-of-variables formula, which handles processes that are neither right- nor left-continuous.

What carries the argument

The load-bearing object is the operator $\mathrm{Pre}$, defined as the first component of the unique solution of the one-barrier predictable reflected BSDE with zero driver. Equivalently, $\mathrm{Pre}[\xi]$ is the predictable Snell envelope of $\xi$: the smallest predictable strong supermartingale dominating $\xi$. The paper rewrites the two-barrier problem as a coupled system $J=\mathrm{Pre}[(\bar J+\tilde\xi^{g,p})1_{[0,T)}]$ and $\bar J=\mathrm{Pre}[(J-\tilde\zeta^{g,p})1_{[0,T)}]$, and solves it by monotone Picard iteration. The Gal'chouk–Lenglart formula provides the a priori estimates that give uniqueness and make the Lipschitz case a contraction in a weighted norm, while Mertens decomposition identifies and makes unique the increasing processes $A,B,A',B'$.

What would settle it

Find a pair of predictable barriers satisfying Mokobodzki's condition for which the Picard sequence (3.15) fails to converge in $S^{{2,p}}$ to a pair solving the coupled system (3.11), or for which the two limits do not satisfy the system; such an example would disprove Lemma 3.2 and hence Theorem 4.1. A concrete check in a simple filtration with one or two predictable jump times would settle it.

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Extended reading notes

Core claim

For any Lipschitz driver $g$ and any pair of predictable admissible barriers $\xi,\zeta$ satisfying Mokobodzki's condition, there is a unique tuple $(Y,Z,M,A,B,A',B')$ in $S^{2,p}\times H^2\times M^{2,\perp}\times (S^{2,p})^2\times (S^{2,p})^2$ solving equation (2.6), respecting $\xi\leq Y\leq \zeta$ at all predictable stopping times, and satisfying the Skorohod minimality conditions (2.7)–(2.8) together with mutual singularity of the pairs $(A,A')$ and $(B,B')$. The pairs deal with the two kinds of jumps: $A$ and $A'$ act only when $Y$ hits a barrier from the left, while $B$ and $B'$ account for right jumps, with $\Delta B=(pY_+-Y)^-$ and $\Delta B'=(pY_+-Y)^+$. Mokobodzki's condition is shown to be necessary for existence in the driver-process case and sufficient under the stated hypotheses. The proof first solves the case where $g$ does not depend on $(y,z)$ by identifying $Y$ with a difference of two predictable strong supermartingales coupled through the one-barrier operator $\mathrm{Pre}$, and then applies a Banach fixed point argument for Lipschitz drivers.

Load-bearing premise

The argument takes as given that the operator Pre from the authors' earlier one-barrier paper exists, is monotone, and maps $S^{{2,p}}$ into itself; the present paper does not reprove or relax that external theorem.

Editorial extensions

If this is right

  • If the driver is a square-integrable process independent of $(y,z)$, the solution exists exactly when Mokobodzki's condition holds, and its first component is given explicitly as $J^p_t-\bar J^p_t+E[\xi_T+\int_t^T g_s\,ds\,|\,\mathcal{F}_{t-}]$.
  • When the lower barrier is right-continuous, the right-jump pushing process $B$ vanishes; when the upper barrier is right-continuous, $B'$ vanishes; and when the barriers are suitably semicontinuous along predictable stopping times, $A$ and $A'$ are continuous.
  • Mokobodzki's condition is a necessary condition for existence, so the result pins down exactly which barrier pairs can support a solution in the predictable setting.
  • The mutual singularity conditions on $(A,A')$ and $(B,B')$ yield uniqueness of the four pushing processes without requiring the usual strict separation $\xi<\zeta$.
  • The contraction argument works in the Banach space $S^{2,p}\times H^2$ with an exponentially weighted norm, so the uniqueness and stability estimates hold uniformly over the whole time horizon.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the theorem is correct, game options with predictable rather than optional exercise times become tractable in filtrations with predictable jumps, a direction the paper's motivation points toward but does not develop.
  • A concrete testable extension would be to write down an explicit two-jump or finite-jump filtration example and verify directly that the solution satisfies $Y=(pY_+\vee \xi)\wedge\zeta$, with the jump sizes of $A,B,A',B'$ matching the stated left- and right-jump formulas.
  • The same coupled-system and contraction machinery may adapt to drivers with jumps or to weaker integrability assumptions on the barriers, though the paper itself does not claim those extensions.
  • The monotone Picard construction suggests that the solution is the minimal pair of predictable strong supermartingales dominating the shifted barriers, which could yield comparison results or a predictable version of the Dynkin-game value beyond what the paper explicitly states.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper introduces a notion of doubly reflected backward stochastic differential equations (DRBSDEs) on a filtered probability space with a general, not necessarily quasi-left-continuous, filtration, where the two barriers are predictable processes. Under a Mokobodzki-type condition, the authors prove existence and uniqueness of a solution (Y,Z,M,A,B,A',B') in S^{2,p} x H^2 x M^{2,perp} x (S^{2,p})^2 x (S^{2,p})^2. The proof proceeds by reducing the (y,z)-independent case to a coupled system of one-barrier predictable RBSDEs solved by Picard iteration, deriving a priori estimates via the Gal'chouk-Lenglart change-of-variable formula, and then using a fixed-point argument for Lipschitz drivers. The main results are Theorem 3.1 and Theorem 3.3 for driver processes and Theorem 4.1 for general Lipschitz drivers.

Significance. If the terminal-jump issue identified below is repaired, the paper is a substantive extension of the DRBSDE theory from the optional/right-continuous setting to predictable obstacles in a general filtration, complementing Grigorova et al. (2018). The use of the Gal'chouk-Lenglart formula to handle both left and right jumps is appropriate, and the a priori estimates in Lemma 3.3 are well suited for the fixed-point step. The proofs are detailed and, apart from the issues listed below and the reliance on the external operator Pre from the authors' prior work [2], internally coherent; the analytic arguments are written in a checkable form. A correct version of the main theorem would be valuable for applications to game options and optimal stopping with non-right-continuous information.

major comments (3)
  1. [Definition 2.6, Eq. (2.6), condition (2.8), Theorem 4.1] The terminal values B_T and B'_T are unconstrained. The equation (2.6) only involves B_{T-} and B'_{T-}, and at tau=T the Skorohod condition (2.8) is automatic because the terminal condition forces Y_T=xi_T. Consequently, with xi=zeta=0 and g=0, the zero tuple is a solution, but so is any tuple with B_t=0 for t<T, B_T=eta, B' identical to 0, and all other components zero, where eta is any non-zero F_{T-}-measurable non-negative square-integrable random variable. This tuple satisfies (2.6), (2.7), (2.8), and dA perp dA', dB perp dB'. Theorem 4.1 therefore asserts uniqueness of an object that is not unique. The gap is readily repaired by requiring B_T=B_{T-} and B'_T=B'_{T-} (or by writing the equation with B_T and B'_T); the existence proof already leaves the value of B_T unused, so the normalization does not affect the constructed solution.
  2. [Section 4, inequality (4.2)] The contraction estimate as printed is miscalculated. From the Lipschitz property of g one obtains |g(t,U_t,V_t)-g(t,U'_t,V'_t)|^2 <= 2K^2(|U_t-U'_t|^2+|V_t-V'_t|^2), so the factor in (4.2) should be 2 K^2 (3+16c^2) (for the sum of the two norms) rather than 2 K (3+16c^2). As written, inequality (4.2) is false for drivers with K>1, although the contraction argument still goes through after replacing K by K^2 and choosing epsilon sufficiently small.
  3. [Definition 2.3] The Lipschitz condition is printed as |g(t,y1,z1)-g(t,y2,z2)| <= K(|y1-y2| - |z1-z2|). The minus sign is a typo; with the minus sign the right-hand side is not a metric and the condition is impossible for non-constant g. The proof of Theorem 4.1 relies on the reversed triangle inequality with a plus sign, so the definition must be corrected to K(|y1-y2| + |z1-z2|).
minor comments (6)
  1. [Section 3, Theorem 3.1 and Lemma 3.3] The space H^{2,p} is used without definition; presumably it is the same as H^2 or a localized version, but it should be defined to avoid ambiguity.
  2. [Section 3.2, proof of Lemma 3.3] The text after equation (3.25) says 'Since beta < 1/epsilon^2', but the lemma assumes beta > 1/epsilon^2; the inequality sign is reversed and should be corrected.
  3. [Remark 2.4] The phrase 'mutually singularity constrain t (2.6)' should refer to condition (iv) of Definition 2.6, not to equation (2.6).
  4. [Proposition 2.1, proof] The phrase 'where the last inequality follows' should be 'where the last equality follows', since the preceding display contains an equality.
  5. [Appendix, proof of Lemma 3.2] The heading 'PROOF OF LEMMA 3.16' is a typo; it should read 'PROOF OF LEMMA 3.2'.
  6. [Throughout] The paper repeatedly uses 'essentially non quasi-left continuous' in the abstract and introduction; the precise assumption (that the filtration is not quasi-left-continuous, or is arbitrary) should be stated unambiguously.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from a distinct one-barrier RBSDE result and a fixed-point argument, not from the conclusion itself.

full rationale

The derivation is self-contained modulo a clearly identified prior theorem. The central construction defines the operator Pre as the first component of the one-barrier predictable RBSDE with driver 0 (Definition 5.2) and imports existence and uniqueness of that operator from the authors' earlier paper [2] (Proposition 5.1). This is a reduction to a different, parameter-free theorem whose assumptions do not include the target DRBSDE; it is used as the base case of a Picard iteration for the coupled system (3.15), not as a restatement of the DRBSDE conclusion. Lemma 3.1 and Proposition 3.1 establish an equivalence between the DRBSDE and an auxiliary coupled system by algebraic decomposition, and the existence proof then solves that system by monotone iteration; the uniqueness proof is a norm estimate based on the Gal'chouk-Lenglart formula plus Mertens decomposition. No equation in the paper is defined in terms of the claimed solution, and no fitted parameter is relabeled as a prediction. The noticeable reliance on the authors' own work is the one-barrier result from [2], but since that is a distinct theorem and not the paper's own conclusion, it is not circular under the stated criteria. A separate question about whether the terminal values B_T and B'_T are pinned down is a mathematical-correctness issue, not a circularity issue.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters fitted to data. Axioms are standard stochastic-calculus theorems plus two domain assumptions: the non-quasi-left-continuous predictable setting and Mokobodzki's condition. The main external reliance is the authors' own one-barrier result [2].

assumptions (6)
  • domain assumption One-barrier predictable RBSDE existence and uniqueness (Bouhadou-Ouknine [2], Theorem 2; Proposition 5.1 here).
    Used to define the operator Pre in Definition 5.2. The coupled system (3.11) and the Picard sequence (3.15) are built on this external result. It is not reproved in this paper.
  • standard math Mertens decomposition for predictable strong supermartingales of class (D).
    Invoked in Lemma 3.1, Equation (3.5), to write Jg,p = N_{t-} - A_t - B_{t-}. Cited from Meyer [22].
  • standard math Gal'chouk-Lenglart change-of-variable formula for optional semimartingales.
    Stated in Theorem 3.2 and used in Corollary 3.1 and Lemma 3.3 for the a priori estimates.
  • standard math Orthogonal martingale decomposition (Jacod-Shiryaev Lemma 4.24).
    Lemma 2.1: any M in M2 decomposes as ∫ Z dW + N with N orthogonal to W. Used throughout the proofs to extract Z and M.
  • domain assumption The filtration satisfies usual conditions and is essentially non-quasi-left-continuous; W is a one-dimensional F-Brownian motion.
    This is the announced predictable setting; the whole paper depends on it.
  • domain assumption Mokobodzki's condition: there exist nonnegative predictable strong supermartingales Hp and H'p with ξ ≤ Hp - H'p ≤ ζ.
    Assumed in Theorems 3.1, 3.3, and 4.1. The paper also proves necessity in Lemma 2.2.

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Pith. "Pith review of Doubly Reflected BSDEs in the predictable setting." pith.science (2026). https://pith.science/paper/5GJOSGOV

@misc{pith2026190808076,
  author       = {Pith},
  title        = {Pith review of: Doubly Reflected BSDEs in the predictable setting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5GJOSGOV}},
  note         = {Machine review of arXiv:1908.08076}
}
read the original abstract

In this paper, we introduce a specific kind of doubly reflected Backward Stochastic Differential Equations (in short DRBSDEs), defined on probability spaces equipped with general filtration that is essentially non quasi-left continuous, where the barriers are assumed to be predictable processes. We call these equations predictable DRBSDEs. Under a general type of Mokobodzki's condition, we show the existence of the solution (in consideration of the driver's nature) through a Picard iteration method and a Banach fixed point theorem. By using an appropriate generalization of It\^o's formula due to Gal'chouk and Lenglart, we provide a suitable a priori estimates which immediately implies the uniqueness of the solution.

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